1972 was the year that the Rolling Stones became, as they loudly proclaimed on tour, the “Greatest Rock ’N’ Roll Band in the World!” Following the violent disaster of their Hell’s Angels–policed Altamont mega-festival outside of San Francisco on December 6, 1969—where three people were killed in accidents and another, Meredith Hunter, was murdered—the Stones disappeared for a while. They went into exile, both real and imagined. Exile from the tax man in England that gazed hungrily on their collected bank accounts and exile as culture warriors—street fighting men—dedicated to the fanciful belief that they could change the world.

The Stones didn’t sit idle in their time out of from the spotlight. They locked themselves away in the sweaty, dank basement of Keith Richards’s mansion in Nellcôte, France, for months on end and came out with their magnum opus: Exile on Main Street. It was a double-album, 18 tracks total, that recounted the near-entire history of blues, country, soul and rock ’n’ roll in the grittiest, most exhilarating way imaginable. Released on May 12, 1972, it sold nearly a million copies in its first week alone.

When plotting their upcoming tour of the United States, the band was intent on making a different impression than they had on their previous jaunt. These shows were going to produce some of the most in-demand tickets of all time, and the Stones and their handlers wanted to be professionals about it. They would do their best to mitigate the violence and mania that seemed to follow their every stop on their last tour, while moving through the country with a degree of style and class that rivaled any visiting foreign dignitary that the U.S. had ever hosted.

Everything about the outing was over the top. The Stones surrounded themselves with the beautiful and the important—Truman Capote, Hugh Hefner, Zsa Zsa Gabor, Woody Allen and Bob Dylan. They heavily courted the press. They blew their minds out with weed, cocaine and tequila, all while staging some of the most incredible sets of live rock music ever seen.

Over 500,000 people mailed in postcards for the mere chance to buy a pass to one of the four shows at Madison Square Garden. Purchases were limited to two per customer at a sticker price of $6.50, but the market was soon flooded with scalped passes that were being hawked at $50 or more.

After months of anticipation, the S.T.P. tour—which, depending on who you asked, either stood for the synthetic hallucinogenic of the same name or simply the Stones Tour Party—kicked off in Vancouver on June 3, 1972. The band and its managers picked this far-flung locale to start their summer swing so that they could knock off the rust before taking their show to more media-saturated markets in San Francisco and Los Angeles. The gig went well. The stop in Seattle went better, and by the time they made it to the Winterland Ballroom in the “City by the Bay,” they were cooking.

Over the next two months, the Rolling Stones and their coterie of crew members and hangers-on crisscrossed North America. At each stop along the way, the local citizenry paused their lives long enough to gawk at, rejoice in or denounce the “event of the year.” In Chicago, the band stayed at Hugh Hefner’s Playboy Mansion when they couldn’t get secure enough hotel rooms elsewhere. In Houston and Fort Worth, Texas, they brought out a Hollywood film crew to capture the show for posterity. Eventually, that footage would get chopped up and released as Ladies and Gentlemen: The Rolling Stones.

A separate, more illicit documentary was also in the works that captured the band’s candid offstage exploits. That film, titled Cocksucker Blues, was produced by Stones manager Marshall Chess and directed by Robert Frank. Though it would never receive an official release, if you search down some of the darker corners of the Internet, it’s not too hard to find. Viewer discretion is advised.

“New York is New York is New York,” famed concert promoter Bill Graham explained to author Robert Greenfield, who tagged along for most of the tour and penned the book A Journey Through America with the Rolling Stones. “’Til you do it there, it hasn’t happened. They could have sold the Garden out for a year. They are the biggest draw in the history of mankind. Only one other guy ever came close—Gandhi.”

New York was ready for the Stones, and the Stones were more than ready to take a bite out of the Big Apple. The band’s four shows spread over three nights at Madison Square Garden were the unquestioned highlights of the tour, with the final night pegged as the can’t-miss gig. Everyone wanted in on the action.

The scene backstage before the last show is a madhouse. Television host Dick Cavett is on hand to interview Mick Jagger and the rest of the band for an hour-long special. This famous face and that famous face drift in and out of view hoping to get a coveted moment of personal time with someone in the band before they make their way out in front of the masses. Few are successful. Makeup is applied, last-minute documents are signed and the tequila sunrises flow like water.

Before the Stones can take the stage at MSG, the audience is treated to a searing set of music from the opening act, Stevie Wonder, who is just then promoting his latest album, Music of My Mind while on the verge of putting out an even greater work, Talking Book. It was a huge coup for the headliners to get the Motown piano virtuoso at this precise moment. His single “Superstition” is one of the biggest songs of 1972, well on its way to claiming the top spot on the charts a few months down the road.

Stevie is the perfect foil for the Stones. He isn’t going to compete with them on their terms like another white blues-based rock band like Humble Pie or the Allman Brothers might. He’s a different thing entirely. His stage show, while rapturous, is centered on his superb musicianship, his incredible songwriting and his own joyful persona.

A little under an hour after Wonder leaves, the Stones make their grand entrance and launch into “Brown Sugar.” Jagger appears dazzling in his sleeveless white jumpsuit, dotted with big sparkling sequins, and a long red sash tethered at the waist. A single, large aquamarine jewel is glued to the center of his forehead. His main foil and “Glimmer Twin” Keith Richards, ever the rock ’n’ roll pirate, is decked out in black leather pants and a flowing, white blouse, which is left unbuttoned to reveal his gleaming white, sweat-glazed chest.

Jagger eats up most of the attention, hopping across the stage, preening like a prized turkey. His arms flail all around, pointing in multiple directions one moment and enticing the crowd to clap along in the next. “Gold coast slave ship bound for cotton fields / Sold in the market down in New Orleans.” The subject matter is chilling, but the thousands singing along sound positively ebullient shouting back each word at their puffy-lipped Messiah.

The pace picks up even more on the next song, “Bitch,” which is played at a far faster clip than the recorded version. By the time they get to “Gimme Shelter” they’re locked in. The shot of adrenaline that comes from getting smacked in the face by 18,000 people simultaneously has worn off, and the Stones settle into their regular groove.

“They had a unique lighting system that they invented,” photographer Bob Gruen told me. The rig was designed by Woodstock MC Chip Monck, who installed a 40-foot by 8-foot array of mirrors near the top of the front of the stage. It was set at a 45-degree angle and had lights shined into it and reflected back onto the band.

Gruen explained, “Up until then in arenas, you had a large spotlight called a Super Trooper, this big carbon arc light up around the rafters in the arena shining down on the stage. They had eight Super Troopers at the back of the stage shining into this Mylar mirror, shining back onto the band. It was the brightest show I’ve ever seen.”

The Stones hit their marks with precision. Keith sounds appropriately cheery during his turn on the microphone for “Happy.” Charlie Watts keeps the party chugging along, crashing cymbals and kicking the hell out of his bass drum on “Bye Bye Johnny.” Mick Taylor sounds like a man possessed during his regular extended solo on “Midnight Rambler,” proving that he is the best pure musician in the band.

Still, it’s Jagger that the fans have come to see, and he doesn’t disappoint. The singer remains the ultimate icon of style and panache. Even with all the obvious effort and energy he expends on stage, he does so in a way that screams cool. When they reach the final song of the main set, “Street Fighting Man,” he’s flinging blood red rose petals over the heads of those screaming their vocal cords to shreds in the front row.

The night isn’t quite over yet. A few moments pass and the band reemerges for a rare encore with Stevie Wonder in tow to perform the latter’s hit “Uptight (Everything’s Alright).” As that song comes to an end, Richards kicks into the instantly recognizable riff to “Satisfaction,” and he and the rest of the band take flight.

The performance is breathtaking. Bobby Keys’s sax mingles with Wonder’s backing band, transforming the straight-ahead rocker into a full, swinging soul sensation. Jagger is giving everything he’s got left in the tank, pushing his voice harder and harder, straining to be heard above the cacophony.

As the last notes of their biggest hit ping across the cavernous walls of the basketball arena, a giant, burning cake is brought onto the stage to commemorate Jagger’s 29th birthday. The crowd is enticed to sing “Happy Birthday.” Somewhere, someone picks up a pie and a full-on food fight takes place in front of thousands of fans. Only Watts seems to be off-limits from the chaos. Once the supply of pies runs out, Mick jumps forward to give the crowd a salute and then he and the rest of the band depart for the last time.

A few hours later, the Stones and their crew, along with a host of the well-to-do, find themselves at the St. Regis Hotel to take part in a final party hosted by Atlantic Records head Ahmet Ertegun. In jarring contrast to the friendly atmosphere inside the Garden, the mood here is almost somber. Everyone in the band are all physically and emotionally spent. The party wraps sometime around dawn and they all scatter to the wind.